Image processing is a highly developed subject of study. More particularly, the description of the geometry of the objects of an image is of great interest for the major application sectors of image processing such as, for example, denoising, improvement of resolution, compression, or image restoration in general.
Ever more complex techniques have been implemented to solve these problems and obtain images of better quality. Numerous studies have thus revealed that the quality of an image is strongly correlated with the quality of the contours.
A logical evolution of the methods has been to render these processing operations adaptive to the content of the image. They thus make it possible to adapt the processing operations in a local manner over certain image areas, by utilizing the homogeneity of certain criteria such as grey levels, texture membership, or contour membership.
The wavelet transform of a signal such as the signal of an image is a mathematical tool used in signal processing. A wavelet is a function comparable with a brief oscillation. It is a mathematical function used to divide a function or a signal in continuous time into various components of scale corresponding to various frequency bands. Each scale component can then be studied at the appropriate resolution.
Carrying out both a frequency decomposition and a temporal decomposition, the wavelet transform presents advantages for representing functions comprising discontinuities and/or narrow spikes with respect to the traditional Fourier transform which performs only a frequency decomposition. The wavelet transform also presents advantages with respect to the Fourier transform for decomposing and recomposing non-periodic or non-stationary, finite signals.
The wavelet transform is a very effective tool for studying the contours of an image by virtue of the separation that it performs between the high and the low frequencies but especially by virtue of the flexibility that it offers in regard to adaptation to resolution.
Transforming an image with the aid of the wavelet transform amounts to separating it into two domains: an approximation domain and a detail domain. The approximation domain comprises the low frequencies of the image corresponding to the continuous components, whereas the detail domain comprises the high frequencies of the image corresponding to the strong variations generally defining a contour.
By virtue of the wavelets, multi-resolution analysis procedures make it possible to represent images in relevant spaces. The information given by multi-resolution approaches is fundamental for the analysis of images, and most particularly for studying the characteristics of the contours.
Theoretical procedures based on geometric wavelets, aimed at detecting the homogeneity of the grey levels along a direction, have been created. Procedures using “bandlets”, “ridgelets” or “curvelets”, or “contourlets” have thus been used.
All these techniques concentrate principally on using the regularity of the wavelet coefficients along certain directions to improve various applications such as denoising or compression for example.
The document WO 2007/059795, the disclosure of which is hereby incorporated by reference, describes a method for improving a signal by using multi-scale grouping bandlets. This procedure makes no choice of resolution, and uses an iterative segmentation which uses significant calculation space and time. The pixel rearrangement procedure used for direction detection in this method exhibits malfunctions entailing the appearance of favored directions.
The document “The curvelet transform for image denoising” written by J. L. Starck et al. and published in the IEEE scientific journal (vol. 11, No. 6, June 2002), the disclosure of which is hereby incorporated by reference, describes the use of “curvelets” which use the “à trous” wavelet transform, and carry out a frequency-based decomposition of the image, but which make no choice of resolution, nor any choice of direction of contours, nor any segmentation as a function of a resolution adapted to detail. The technique described in this document is adapted only to certain types of image processing operations such as noise reduction. On the other hand, this technique is not adapted to image processing operations of encoding or interpolation type, because it allows neither a choice of resolution, nor a choice of direction.
Certain geometric flow creation techniques not using wavelets have also been studied previously. Thus, in the document “Orientation diffusions” drafted by P. Perona and published in the IEEE scientific journal (vol. 7, No. 3, March 1998), the disclosure of which is hereby incorporated by reference, is described an approach which uses grey levels rather than the wavelet transform. The iterative procedure described is based on the diffusion of the orientations of the gradients of an image. This approach does not make it possible to get a global picture of the contour, and the discontinuities remain difficult to detect.